Is there an obligation to put them on every day, or is it simply that whoever does more is especially praiseworthy?
For example, is it preferable to put them on for two hours on Sunday and not at all on Monday, or is it better to put them on for one minute on Sunday and one minute on Monday?
If the second option is correct, then what is the source?

That’s a new question.
What is accepted among the halakhic decisors is that there is an obligation to put them on every day, and also that whoever does more is especially praiseworthy. In other words, option 2 is preferable.
However, this rule—that tefillin obligate one every day—does not have a clear source. From the wording of Maimonides, chapter 4, halakha 26, it appears that their commandment applies all day, and nothing is written there about every day. True, it comes out as every day, because one must constantly exclude nights, but there is no law of a commandment tied to the day as such. And see also Tosafot on Sukkah 45b.
In fact, the Levush, Orach Chayim 25:12, wrote that their commandment applies every day, and it requires investigation what his source is.
An interesting practical difference comes up in the responsa Lehorot Natan, vol. 11, sec. 5, regarding someone who crossed the International Date Line. There the date changes, but there is no renewal of a new day in the sense of sunrise, and there it is obvious to him that one does not need to put on tefillin again—because of the definition that their commandment applies all day and not “every day.”
I later saw that this was discussed in an article in Neveh Heikhal, Parashat Bo, 5777 (it’s supposed to be online). There he proved from several medieval authorities who argued that tefillin is a positive time-bound commandment—from which we derive women’s exemption—because it does not apply on the Sabbath, on Jewish holidays, or at night. And it is clear that were it not for that, women would be obligated; meaning, the commandment does not renew itself each day.