Le sens que vous recherchez peut être celui de titi parisien.

C’est le mot familièrement employé pour désigner un jeune garçon malicieux (faux ami : aucune malveillance, mais de l’espièglerie) et jovial, qu’il soit de Paris ou d’ailleurs …

Ajout suggéré par Romain
… ou une référence au dessin animé “Titi et Gros minet”, Titi étant canari sans cesse pourchassé par Gros minet.

Titi is colloquial French – originally (19th century) applied to a young apprentice it is nowadays (although it seems the word is getting out of use among the younger generations) a smart and spirited young boy.
Gavroche (Victor Hugo’s character in Les Misérables) personifies the “Titi parisien” but you don’t need to be Parisian to be called a titi ; the word brings to mind a youngish boy, rather slim, not very tall and unpretentious.

It is the kind of word that is not translatable into another language, it needs to be explained. I wanted to see how some dictionaries dealt with it and I hit upon “urchin”, I entirely disagree with that translation: urchin implies mischievousness: there’s no mischievousness in a titi, just high-spiritedness.

Indeed your friend does not mean titi in the sense of a kind or tree or monkey (which it can also mean) and she might ignore these two meanings of the word.

Titi has a few significations.

It can mean Titi parisien, like the other answers described thoroughly, some kind of playful young rascal boy.

It is also the name of the bird “Tweety” in the Looney Toons.

It can finally be a simple nick name based on your name. It often applies to boys named Thierry or to girls named Leticia, for instance. It is not limited to theses names and it is also applied to people who have the syllabe it in their patronym, which seems to be your case.

When French people make nicknames, they often take a syllable (or a part) of the real name and repeat it twice.

Because many words in French with twice the same syllable sound cute (especially for girls), including a part of the baby vocabulary (i.e. bébé = baby, mimi=mignon=cute, tata=tante=aunty).

I have lived in Paris since I was born and I do not know the word “titi parisien”. So even if it exists, I am pretty sure that it is not what your friend meant.

We French use it quite frequently. It is actually an abbreviation of the word “petit” or “petite”. It is not an insult but rather a term of endearment.