There have been several attempts to point to alleged homosexual couples in the Bible, primarily Ruth and Naomi, Daniel and Ashpenaz, and David and Jonathon. The arguments regarding Ruth/Naomi and Daniel/Ashpenaz are far from compelling for me. The arguments regarding David and Jonathon, however, while not quite compelling, leave open the strong possibility that they were involved in an homosexual marriage. 

Starting from the crux of the argument at 1 Samuel 18:21, Saul tells David, that by marrying Saul's daughter Michal, David will be his son-in-law for the second time (Hebrew: "bstym ttctn by hynm"). The actual translation of this phrase is somewhat controversial, being literally translated "You will become my son-in-law through two." In this instance, the correct interpretation of this verse is crucial, because it radically shapes our view of David and Jonathon's relationship, since Scripture only indicates that David had any kind of relationship with two of Saul's children: Jonathon and Michal. Some translations interpret this verse as meaning that Saul "said for the second time," or that David has a "second opportunity" to become Saul's son-in-law. These interpretations, however, are strained, and the Hebrew does not easily lend itself to mean either of these. Most standard translations clearly interpret the verse to mean that David will become Saul's son-in-law for the second time (NIV being the primary exception, and the RSV is ambiguous): 
 

ASV: Wherefore Saul said to David, Thou shalt this day be my son-in-law a second time.  RSV: Therefore Saul said to David a second time, "You shall now be my son-in-law."  BBE: So Saul said to David, Today you are to become my son-in-law for the second time.  DBY: And Saul said to David, Thou shalt this day be my son-in-law a second time.  YLT: Saul saith unto David, `By the second -- thou dost become my son-in-law to-day.'  NAS: Therefore Saul said to David, "For a second time you may be my son-in-law today."

The question then becomes what Saul actually meant if he is telling David that he will become his son-in-law for the second time. The first offer Saul made to David for a wife was Merab, but she married Adriel of Meholah instead (18:19). The only other covenant made between Saul's family and David was between David and Jonathon in 18:3, which is not a covenant of business or politics, but of friendship/love ("ahbh"). Moreover, this relationship is described in very strong emotive language, starting in 18:1. Prior to looking at this more closely, an understanding of the story up to this point is helpful. In chapter 17, we find David's older brothers going to war against the Philistines while David stays at home. David is then sent to take food to his brothers, following which is the classic David and Goliath story. As David goes back to Saul after killing Goliath, we see that David is totally unknown to King Saul (17:58). However, as David talks to King Saul, Jonathon falls in love with David, after having never met him, or talked to him (which has a vague sound of "love at first sight" in our culture). 
 

1 Samuel 18:1-4 (NIV)  1 After David had finished talking with Saul, Jonathan became one in spirit with David, and he loved him as himself.  2 From that day Saul kept David with him and did not let him return to his father's house.   3 And Jonathan made a covenant with David because he loved him as himself.   4 Jonathan took off the robe he was wearing and gave it to David, along with his tunic, and even his sword, his bow and his belt.

While there is no similarity between the Hebrew phrases in 1 Samuel 18:1-2 above and in Genesis 2:24, there is a striking similarity in concepts between the son leaving the parents to join to a spouse, and the two becoming one: 
 

Genesis 2:24 (NIV)  For this reason a man will leave his father and mother and be united to his wife, and they will become one flesh.

When we put together chapter 18, from the beginning, with Jonathon's strong emotional affection towards David and their subsequent covenant, to the end, where we see Saul referring to David being his son-in-law a second time with his marriage to Michal, we see the very strong possibility that David and Jonathon were joined in a covenant that Saul recognized as a marriage. This line of reasoning, while persuasive to me, it is not conclusive. First, I don't know that we have any other extant Hebrew literature of that era that refers to a gay marriage, which would lead one to question whether or not Saul would have seen David and Jonathon's covenant as one of legal marriage. If not, then the only possibility for Saul's language in 18:21 is that he was referring to David's second son-in-law status as coming from the original promise by Saul to give Merab to him (18:17), even though Merab married another man. A second possible criticism is that this argument is made from conjecture, that no specific reference is made to marriage (ynh, yqch) or sexual activity. This, however, is not a valid criticism. The words referring to marriage in the Old Testament are typically in the context of being "taken" or "given" (yqch) as property (byvlh) or protector/provider (ybm), since women had no rights in Hebrew culture, and were considered property to be given/sold. This aspect of marriage would not have been applicable to David and Jonathon's relationship. The other primary word translated as marry is actually the exact same word as "woman" (ishh), which obviously isn't applicable in this case. As for the lack of specific reference to sexual activity which would definitively signify marriage, very few Old Testament relationships which are clearly marriage relationships have subsequent descriptions of sexual activity, therefore it is improbable that such a characterization would be applied here either. However, 2 Samuel 1:26 may even be a reference to sexual activity between David and Jonathon. After Jonathon has been killed, David mourns his death, and says the following (NIV): 
 

26 I grieve for you, Jonathan my brother; you were very dear to me. Your love for me was wonderful, more wonderful than that of women.

In Hebrew culture, similar to many mid-eastern cultures today, men and women did not engage in platonic relationships. They were either married, or they had no relationship. In this case, David compares his relationship with Jonathon to the relationship with a woman, strongly indicating a marriage/sexual relationship. Further, the word used for love here (ahbh; used also in 1 Samuel 18:3 and 1 Samuel 20:17 referring for Jonathon's love for David) is the same word used in Genesis 29:20 for Jacob's love for Rachel, and is used repeatedly in Song of Songs. It is typically translated as love in the context of a marriage or sexual desire (Proverbs 5:19, etc.; see Strong's concordance #0160). 

After this analysis we are left with two questions. First, could Saul have legally seen David and Jonathon's covenant as marriage, to the extent that he would call David a son-in-law. Second, is the intensity of the language referring to Jonathon's love and covenant with David, and David's reference to his love for Jonathan, enough to sustain the belief that they were engaged in a marriage covenant? Neither of these questions can be answered definitively. Whether or not Saul would have legally condoned this relationship can only be answered with further research into the marriage documents from that time, which I don't know exist anymore. However, the conceptual parallel of marriage between 1 Samuel 18:1-2 and Genesis 2:24, the intensity and type of language used in 1 Samuel 18:1-4 and subsequent covenant between Jonathon and David, and David's comparison of his love to that of women certainly leads me to the conclusion that their relationship could have been one of marriage.